0001000101000emaxminarea.ACT0007020011construction1.EAC010000002e5c^[\ animationdQa)`PYdQa)tvrALCuxy5@  H  A@6$(Rc  (9-x^2)^(1/2)Du! @  !B@6 ! @6` H  D@6  @    @   @6f#$tY ` @   : @    @6v#(  $(Rd`H! F@6$(Rd@  @    @6`@  @    @6`!`E@6@  @  @    @6 ywR#Y`$(Rd!@  W2g@  @    @6  !C@6!@  "@  #@  $! @6 Y ywR"   %!@  $W2g&@  $$"!  $  [ Construction:[1. Insert 'Geometry' strip[2. Draw an ellipse using[ function [y=9-x2.[3. Select a point on the[ellipse.[4. Construct two points[B and C.[5. Construct a perpendi-[cular line from A to BC.[6. Construct the point D.[7. Draw a point E.[8. Construct a perpendi-[cular line from A to BE.[9. 'Construct' the point[F.[10. Construct line [segments AF,FB,BD and [DA.[**11. Add animation A[on the ellipse.[ Edit Animate:[t0: from 0 to 3 step 30.[030 0.1034482759 2.998215879 0.3101602633 0.2068965517 2.992857133 0.6192118206 0.3103448276 2.983904504 0.9260393288 0.4137931034 2.971325507 1.229514003 0.5172413793 2.955073833 1.528486465 0.6206896552 2.935088474 1.821779053 0.724137931 2.911292541 2.108177357 0.8275862069 2.883591696 2.386420714 0.9310344828 2.851872156 2.655191317 1.034482759 2.815998122 2.913101505 1.137931034 2.775808524 3.158678665 1.24137931 2.731112852 3.390346988 1.344827586 2.681685806 3.60640505 1.448275862 2.627260365 3.804997771 1.551724138 2.567518685 3.984080719 1.655172414 2.502079991 4.141373778 1.75862069 2.430484164 4.274299736 1.862068966 2.352169035 4.379900961 1.965517241 2.266438169 4.454723298 2.068965517 2.172413793 4.494649227 2.172413793 2.068965517 4.494649227 2.275862069 1.954597617 4.448394576 2.379310345 1.827260869 4.347620688 2.482758621 1.684015924 4.181005052 2.586206897 1.520372944 3.931998993 2.689655172 1.328817163 3.574059956 2.793103448 1.094793646 3.057871908 2.896551724 0.7810173554 2.26225716732.449598621-67.348797257-6[\list'PGraph2DT Graph3Dh LISTSYSt4NModify $ STATCALC NSTATSYS \NSequence0, Sheet\| Sheet3D| SolveEqTSolveLwrX SolveUprd StupFLG1p(StupListD StupPict$ ViewWind  ( 4 @ L X d p |          ! " # $ % &$ '0 (< )H *T +` ,l -x . 0 1 2 3 4 5 E F H I J K L M, N8 OD PP Q\ Rh St T ] ^p _T `8 a< b@ dD e(   $ 0 < system]listsystem^]=system_^system`_systema`systemb~ aseq_histbNewFolde system]listsystem^]=systemsystem]listsystem_]=systemsystem]listsystem^]=systemsystem]listsystem^]=systemsystem]listsystem^]=systemsystem]listsystem^]=systemsystem]listsystem^]=systemsystem]listsystem^]=systemsystem]listsystem^]=systemsystem]listsystem^]=system  0@PSheet1ju^Sheet2ju^Sheet3ju^Sheet4ju^Sheet5ju^SheetSheet3D 0@PSheet1ju^Sheet2ju^Sheet3ju^Sheet4ju^Sheet5ju^SheetSheet3D i ^ $ `P`P`Y03vb3vb E@Y95y$(10qy`#YuY $y0IFI"p $0<HT`lx ,8DP\Du eQp 4H'` y1@ $y0  hU $y1 'Xb 1D Duy1@$y14H'X`Du U$eQrA@u i h`U$eQp$y0'Xb74PH'XbXbphU y1DeQr@ $0<HT`lx ,8DP\(W09P@%PpPs0PG@T5i`r`Y wXR@s h`br`6PVuhPP yC@5!iP&d8$y0eQpEapr`h@@Rr@20 Gd`sU@ Db  $0<HT`lx ,8DP\c0 ! ` &( "0RFPy0w5p8d q@eQ1p1PPxfP9F`dIw@qsw'Bs`7EG#)IFI"pIFI"pDW`4v h 0W@Y`xq&"Wp4%p  $0<HT`lx ,8DP\Du eQp 4H'` y1@ $y0  hU $y1 'Xb 1D Duy1@$y14H'X`Du U$eQrA@u i h`U$eQp$y0'Xb74PH'XbXbphU y1DeQr@ $0<HT`lx ,8DP\c0 ! ` &( "0RFPy0w5p8d q@eQ1p1PPxfP9F`dIw@qsw'Bs`7EG#)IFI"pIFI"pDW`4v h 0W@Y`xq&"Wp4%p `'q Q[eAct020011construction2.EAC010000002e3d^[ Construction.\ A rectangle``vrALCuxy  H  F@6 X `i  D@6H  E@6B21DHH  C@6B21DH X `i 5H   ! @6qI7Su  `deH `Q'`  `A@6  B@6 @  @  )iC# &1WG7 @    @6` X `i @  @    @6 YB21DH@  H!   @6@  @    @6 @  @    @6 @  @  @    @6r%hFTB  X `iY @  ! ! @6 "@  !#@  !oJ$@  %  @6`r%hFTB B21DH&!H  W2g@'@  (!H  !%W2g@)@  %%#"! !%[1. Draw a line segment[AB; be sure A is on the[y-axis and B is on the x-[axis.[2. Pick a point C on AB; [be sure C is on AB.[3. Draw a line segment[DB and DA; be sure[D=(0,0).[4. Draw perpendicular[lines from C to DB and[ from C to DA [ respectively.[00 0.1473684211 0.2722437673 0.2947368421 0.5142382271 0.4421052632 0.7259833795 0.5894736842 0.9074792244 0.7368421053 1.058725762 0.8842105263 1.179722992 1.031578947 1.270470914 1.178947368 1.330969529 1.326315789 1.361218837 1.473684211 1.361218837 1.621052632 1.330969529 1.768421053 1.270470914 1.915789474 1.179722992 2.063157895 1.058725762 2.210526316 0.9074792244 2.357894737 0.7259833795 2.505263158 0.5142382271 2.652631579 0.27224376732.80[\*``vrALCuxy+!@  , ! @6cT$e&Y 1WvHI 6UVF3`-!`A@6P` .!`B@6E` /@  -,0@  .,(1 @6 1WvHPYcT$e#Y86T`W2g2@  3H F@6x 20(`'SRwX110/,42H! 5H E@6x!ygit`'SRwX`36@  37! @6E`  hg.-8%@  .79@  -7:@  3;€! @6P`  hg-.<%H  -;=@  .;>@  57?@  5;35@@  A `D@6`YB  @6DWEx'Ydp 9SyF0C!`C@6pp`AD@  CB@4DBAC897<=;.--.;7CAB53,1\Analyze the dataEGraph2D Graph3D LISTSYS4NModify 4$ STATCALC XNSTATSYS `\NSequence, Sheet| Sheet3Dd| SolveEqSolveLwr SolveUpr StupFLG1(StupList$D StupPicth$ ViewWind y1LHy2H  $ 0 < H T ` l x         ! " # $ % & ' (, )8 *D +P ,\ -h .t 0 1 2 3 4 5 E F H I J K L M N( O4 P@ QL RX Sd Tp ]|D ^D _ ` a  b dD eXD           system]listsystem^]=system_^system`_systema`systemb~ aseq_histbNewFolde  system]listsystem^]=system1system]listsystem^]=systemsystem]listsystem^]=systemsystem]listsystem^]=systemsystem]listsystem^]=systemsystem]listsystem^]=systemsystem]listsystem^]=systemsystem]listsystem^]=systemsystem]listsystem^]=systemsystem]listsystem^]=system  0@PSheet1ju^Sheet2ju^Sheet3ju^Sheet4ju^Sheet5ju^SheetSheet3D 0@PSheet1ju^Sheet2ju^Sheet3ju^Sheet4ju^Sheet5ju^SheetSheet3D i ^ $ < 3x-0.6964285716022*x^2+1.950000000568*x-1.700000140659-10 3xA(x)`P`P`Y r$7g@Yc4b`D(10qy`#YuY  $0<HT`lx G6! shB BRc G6 6!0 !&0 xpG62cxG6!bRc v!0WG@1WP!&1`5xspPRce&1W $0<HT`lx r$7g0 #' %3yP G$@ %v " 'p@3 iR6p6p3 iR'p@" %v  G$@ %3yP #' r$7g0  $0<HT`lxG6! shB BRc G6 6!0 !&0 xpG62cxG6!bRc v!0WG@1WP!&1`5xspPRce&1W $0<HT`lxr$7g0 #' %3yP G$@ %v " 'p@3 iR6p6p3 iR'p@" %v  G$@ %3yP #' r$7g0 Bq`"YhpYYd"I `Pu89G X[[We define the area of [the rectangle as followsRdefine A(x)=x*(-0.6964x+1.95)RdoneR expand(A(x))R1.95x-0.6964x2[ Exercise.[1) How do we find the[maximum of A(x)?[eActAH(xx*(-0.6964*x+1.95)02000bellipse.EAC010000003dbb^*[Largest area and [ellipse[\AuthorProfessor,Dr.Wei-Chi Yang Department of Math/Stats Radford University Radford, VA 24142 USA e-mail: wyang@radford.edu URL:www.radford.edu/ ~wyang[ Objective.[Find the largest [rectangle in the first [quadrant under an [ellipse.[ Exploration.[Find the largest area of[the rectangle (in the [1st quadrant) under [y=((36-9x^2)/4)^(1/2).[[\ Animation``vrALCuxy  H  A@60'b06H  D@60!`F@6H  E@6'b06 5@!    ((36-9x^2)/4)^(1/2)0&1WG !@  ! @6 Y @   @    ! @6 Y0@  @  @   ! @6`'b06@  @   !! @6@   @    @6 `B@6`H  C@6@  @  @    @6Q1EwY`'b06 !@  !" @6`r%hFTB 0W2g"@  #@  $@  !%!@  !W2g@&@  !!  !  [00 0.1052631579 0.3153517888 0.2105263158 0.6280701483 0.3157894737 0.9354845837 0.4210526316 1.234848241 0.5263157895 1.52329386 0.6315789474 1.797781799 0.7368421053 2.055035769 0.8421052632 2.291459892 0.9473684211 2.503027226 1.052631579 2.685123727 1.157894737 2.832320231 1.263157895 2.938022798 1.368421053 2.993904997 1.473684211 2.988918384 1.578947368 2.907399837 1.684210526 2.724950896 1.789473684 2.397503325 1.894736842 1.819773832 1.894736842 1.819773832 [list1=x value, [list2=y value=area \ drag and drop'Graph2DT Graph3Dh LISTSYSt4NModify $ STATCALC NSTATSYS \NSequence0, Sheet\| Sheet3D| SolveEqTSolveLwrX SolveUprd StupFLG1p(StupListD StupPict$ ViewWind  ( 4 @ L X d p |          ! " # $ % &$ '0 (< )H *T +` ,l -x . 0 1 2 3 4 5 E F H I J K L M, N8 OD PP Q\ Rh St T ]D ^D _ ` a b  d$D ehD      system]listsystem^]=system_^system`_systema`systemb~ aseq_histbNewFolde system]listsystem^]=systemsystem]listsystem^]=systemsystem]listsystem^]=systemsystem]listsystem^]=systemsystem]listsystem^]=systemsystem]listsystem^]=systemsystem]listsystem^]=systemsystem]listsystem^]=systemsystem]listsystem^]=systemsystem]listsystem^]=system  0@PSheet1 ^Sheet2 ^Sheet3 ^Sheet4 ^Sheet5 ^SheetSheet3D 0@PSheet1 ^Sheet2 ^Sheet3 ^Sheet4 ^Sheet5 ^SheetSheet3D i ^ $ `P`P`G6 YBRbGd1B x @YY&d(10qy`#YuY G6 s  $0<HT`lx &1W Rc xsp !&1` &1WP 1WG@ 6!0 BRc G6! &1Wxsp&1WP6!0G6!WG6hBR`xsh@G6 G6  $0<HT`lx 5 (H0 5HEp #HH$R2ywyP5v)Y P0'"`hQ#rp# #"y9p8@sprIP`9u2Ps s  $0<HT`lx&1W Rc xsp !&1` &1WP 1WG@ 6!0 BRc G6! &1Wxsp&1WP6!0G6!WG6hBR`xsh@G6 G6  $0<HT`lx5 (H0 5HEp #HH$R2ywyP5v)Y P0'"`hQ#rp# #"y9p8@sprIP`9u2Ps s `QP#h Q[[By using the 'trace' from[the scatter plot above,[we will get a rough idea[where the maximum area[ should be.[[ Analytically.[We can set up the area[function as follows:Rdefine A(x)=x*36-9x2RdoneRA(x)xR-18x2-36-9x2+36Rrfactor(A(x)xR-18x-2x+2-9x2+36R\plot A and A' (HGraph2DT Graph3Dh LISTSYSt4NModify $ STATCALC NSTATSYS \NSequence0, Sheet\| Sheet3D| SolveEqTSolveLwrX SolveUprd StupFLG1p(StupListD StupPict$ ViewWind y1HHy2dH|              $ 0  < !H "T #` $l %x & ' ( ) * + , - . 0 1 2 3 4 5, E8 FD HP I\ Jh Kt L M N O P Q R S T ] ^ _ ` a b    ( 4 system]listsystem^]=system_^system`_systema`systemb~ aseq_histbNewFolde system]listsystem^]=systemsystem]listsystem^]=systemsystem]listsystem^]=systemsystem]listsystem^]=systemsystem]listsystem^]=systemsystem]listsystem^]=systemsystem]listsystem^]=systemsystem]listsystem^]=systemsystem]listsystem^]=systemsystem]listsystem^]=system  0@PSheet1ju^Sheet2ju^Sheet3ju^Sheet4ju^Sheet5ju^SheetSheet3D 0@PSheet1ju^Sheet2ju^Sheet3ju^Sheet4ju^Sheet5ju^SheetSheet3D i ^ $ 6 3x(-18*(x-(2)^(1/2))*(x+(2)^(1/2)))/(-9*x^2+36)^(1/2) 3xA(x)`P`P`p`p Th7WW`(10qy`#YuY ABV#s`GypI Q[It is easy to see that[the max of A(x) occurs[at x=2.eActAH$xx*(36-9*x^(2))02000cparabola.EAC010000002a0b^([Find the largest rectangle[in the first quadrant under [ the parabola,[y=4-x^2[\ AnimationP`y$P`Iy$PvrALCuxy  H  A@601H  D@60!`F@6H  E@615@!    4-x^20&1WG !@  ! @6 Y @   @    ! @6 Y0@  @  @   ! @6`1@  @   !! @6@   @    @6 `B@6`H  C@6@  @  @    @6Q1EwY`1!@  " @6`r%hFTB 0W2g!@  "@  #@   $!@   W2g@%@       [-The 1st column is the [x value.[-The 2nd column is the [y coordinate = areas[00 0.1052631579 0.4198862808 0.2105263158 0.8327744569 0.3157894737 1.231666424 0.4210526316 1.609564076 0.5263157895 1.95946931 0.6315789474 2.274384021 0.7368421053 2.547310104 0.8421052632 2.771249453 0.9473684211 2.939203966 1.052631579 3.044175536 1.157894737 3.079166059 1.263157895 3.037177431 1.368421053 2.911211547 1.473684211 2.694270302 1.578947368 2.379355591 1.684210526 1.95946931 1.789473684 1.427613355 1.894736842 0.776789619520[\Import data to a list&Graph2D Graph3D0 LISTSYS<4NModify p$ STATCALC NSTATSYS \NSequence, Sheet$| Sheet3D| SolveEqSolveLwr SolveUpr, StupFLG18(StupList`D StupPict$ ViewWind y1|Hy2`Hx               ,  8 !D "P #\ $h %t & ' ( ) * + , - . / 0 1 2 3 4 5 E( F4 H@ IL JX Kd Lp M| N O P Q R S T ]D ^ D _d `h al bp dtD eD     , 8 D P \ h t  system]listsystem^]=system_^system`_systema`systemb~ aseq_histbNewFolde  system]listsystem^]=system1system]listsystem^]=systemsystem]listsystem^]=systemsystem]listsystem^]=systemsystem]listsystem^]=systemsystem]listsystem^]=systemsystem]listsystem^]=systemsystem]listsystem^]=systemsystem]listsystem^]=systemsystem]listsystem^]=system  0@PSheet1ju^Sheet2ju^Sheet3ju^Sheet4ju^Sheet5ju^SheetSheet3D 0@PSheet1ju^Sheet2ju^Sheet3ju^Sheet4ju^Sheet5ju^SheetSheet3D i ^ $ i 3x-5.894936508018-10*x^4-0.9999999972165*x^3-4.820825655402-9*x^2+4.000000003076*x-4.7338337185-10 3xA(x)`P`P`Y UAUB 2YiI'(10qy`#YuY  $0<HT`lx@"qXYP`P`yv``&`` $0<HT`lx &1W Rc xsp !&1` &1WP 1WG@ 6!0 BRc G6! &1Wxsp&1WP6!0G6!WG6hBR`xsh@G6   $0<HT`lx b 2wDV #fB@`d`i1'CTs@wIE0`AuS`fqwCTpiBp0 7UYi1Bv5PvxP  $0<HT`lx&1W Rc xsp !&1` &1WP 1WG@ 6!0 BRc G6! &1Wxsp&1WP6!0G6!WG6hBR`xsh@G6  $0<HT`lxb 2wDV #fB@`d`i1'CTs@wIE0`AuS`fqwCTpiBp0 7UYi1Bv5PvxP I6PY !eY%eTY0vs83qYUS#@x `g!H4! [[After getting the scatter[plot for list 2(x value)[and list 1(y value for [area), we can trace [where the maximum could[be. In this case, it is[around x=1.1478947.[[**Analytically, [we can set the area R function and Rdefine A(x)=x*(4-x^2)Rdone[It is easy to see that[A(x) is a cubic function[which we can approximate[its maximum or use its[derivative to find the [maximum, which we will[ omit here, [ the important[thing is to see the [cubic regression of [the list2-list1 is[ identical to [y=A(x).[eActAH xx*(4-x^2)02000drectangle.EAC010000002812^[Maximum area of the[rectangle under a line,[x-axis and y-axis.\ An animation``vrALCuxy   D@6H  F@6S(SbbH  C@6)FAdFS(SbbH  E@6)FAdF5H   ! @6dV %dFAP i%qQPw`  `A@6  B@6 @  @  1TpUH &1WG7 @   ! @6`@  @   !! @6)FAdF`@  @   ! @6S(Sbb`@  u@   !! @6`@  oJ@    @6` @  @  @  ! @6r%hFTBY`S(Sbb @  ! ! @6  "@  !#@  !oJ$@  %  @6 Y ywR# )FAdF&!H  W2g@'@  (!H  !%W2g)@  %%#"! !%[-first column=x values[-2nd column= areas[00 0.1578947368 0.3066481994 0.3157894737 0.5792243767 0.4736842105 0.8177285319 0.6315789474 1.022160665 0.7894736842 1.192520776 0.9473684211 1.328808864 1.105263158 1.431024931 1.263157895 1.499168975 1.421052632 1.533240997 1.578947368 1.533240997 1.736842105 1.499168975 1.894736842 1.431024931 2.052631579 1.328808864 2.210526316 1.192520776 2.368421053 1.022160665 2.526315789 0.8177285319 2.684210526 0.5792243767 2.842105263 0.306648199430[\ copy to list*PGraph2D Graph3D LISTSYS4NModify H$ STATCALC lNSTATSYS t\NSequence, Sheet| Sheet3Dx| SolveEqSolveLwr SolveUpr StupFLG1(StupList8D StupPict|$ ViewWind y1LHy2Hy3 H@ L X d p |             ! " #$ $0 %< &H 'T (` )l *x + , - . 0 1 2 3 4 5 E F H I J, K8 LD MP N\ Oh Pt Q R S T ]D ^D _8 `< a@ bD dHD eD        $ 0 < system]listsystem^]=system_^system`_systema`systemb~ aseq_histbNewFolde  system]listsystem^]=system1system]listsystem^]=systemsystem]listsystem^]=systemsystem]listsystem^]=systemsystem]listsystem^]=systemsystem]listsystem^]=systemsystem]listsystem^]=systemsystem]listsystem^]=systemsystem]listsystem^]=systemsystem]listsystem^]=system  0@PSheet1ju^Sheet2ju^Sheet3ju^Sheet4ju^Sheet5ju^SheetSheet3D 0@PSheet1ju^Sheet2ju^Sheet3ju^Sheet4ju^Sheet5ju^SheetSheet3D i ^ $ < 3x-0.6833333333704*x^2+2.050000000036*x+2.600009096909-11 3xA(x) 3x:(A(x),x,1)`P`P`Y03vb3vb d@Yd(10qy`#YuY  $0<HT`lx WG6 xsp shBP 1WG@ G6 G6! Rc&1WPBRc WG6shBPG6 &1W!&1`6!0RcxhBR`!&0  $0<HT`lx d@ y"Cvp r1 !`fP% w`2@C$IhPS2@pS2@pIhPC$2@% w`!`fPr1 y"Cvp d@  $0<HT`lxWG6 xsp shBP 1WG@ G6 G6! Rc&1WPBRc WG6shBPG6 &1W!&1`6!0RcxhBR`!&0 $0<HT`lxd@ y"Cvp r1 !`fP% w`2@C$IhPS2@pS2@pIhPC$2@% w`!`fPr1 y"Cvp d@ 3337Y6` i YP `C1c Y[[Analytical Approach.[ Area=x*y=x*y=x*(-0.6833x+2.05)Rdefine A(x)=x*(-0.6833x+2.05)RdoneRsimplify(A(x))R-6833x-20500x10000R diff(A(x),x)R-6833x-102505000ReActAH(xx*(-0.6833*x+2.05)02000esemicircle.EAC010000002461^ [Find the largest rectangle[in the first quadrant under [ a semicircle.[\ Animation``vrALCuxy  H  A@60QhASWH  D@60!`F@6H  E@6QhASW5@!    (4-x^2)^(1/2)0&1WG !@  ! @6` @   @    ! @6`0@  @  @   ! @6`QhASW@  @   !! @6@   @    @6 `B@6`H  C@6@  @  @    @6Q1EwY`QhASW!@  " @6`r%hFTB 0W2g!@  "@  #@   $!@   W2g@%@       [00 0.1052631579 0.2102345259 0.2105263158 0.4187134322 0.3157894737 0.6236563891 0.4210526316 0.8232321605 0.5263157895 1.01552924 0.6315789474 1.1985212 0.7368421053 1.370023846 0.8421052632 1.527639928 0.9473684211 1.668684817 1.052631579 1.790082484 1.157894737 1.888213487 1.263157895 1.958681865 1.368421053 1.995936665 1.473684211 1.992612256 1.578947368 1.938266558 1.684210526 1.816633931 1.789473684 1.59833555 1.894736842 1.213182555 1.894736842 1.213182555[-first column = x values[-2nd column= areas[\ list editor&Graph2DT Graph3Dh LISTSYSt4NModify $ STATCALC NSTATSYS \NSequence0, Sheet\| Sheet3D| SolveEqTSolveLwrX SolveUprd StupFLG1p(StupListD StupPict$ ViewWind  ( 4 @ L X d p |          ! 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